This chapter presents the design of the experiments that were run in order to recreate the findings of Ruxton and Saravia\cite{RuxtonSaravia} as well as attempting to determine whether there is any difference to the outcome of the system whether a cellular automaton or a Gillespie simulator is used. It provides the settings for each simulation and the measurements for all of the experiments.

The overall goal of this project is to determine whether there is any difference between using a discrete step cellular automaton and a Gillespie simulator. This will be done by first verifying the results of the research completed by Ruxton and Saravia\cite{RuxtonSaravia} and adding into it the comparison with a Gillespie simulator. The experiments will then be continued to determine the effects, if any, of introducing a second species and to modify the birth and death probabilities of each species as well.

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\section{Replication of Ruxton and Saravia with added Gillespie Simulator}
\label{sec:exp.rux}

Following the paper published by Ruxton and Saravia\cite{RuxtonSaravia}, I carried out the same experiments in order to verify their results and also compared the Gillespie simulator against them. All the experiments were carried out using the discrete step rules SdF, SdC, RdF and RsR all with both time types 1 and 2 as well as the Gillespie simulator.

The purpose of this experiment is to verify that there are notable differences on the system based on the different update rules, and also to compare a sample of different cellular automaton rules to the Gillespie simulator to look for any correlations.

\subsection{Extinction Times}
\label{sec:sec:exp.rux.ext}
Ruxton and Saravia first calculated the Critical Extinction Probabilities (CP$_{e}$) of each setting using the method of Buttell et al\cite{Buttell} and then chose a death probability higher than the CP$_{e}$ for each setting to run in their experiments. For my simulations, I did not repeat this calculation choosing instead to use the death probability (0.6) that they determined.

To determine the Extinction Median (the median for the number of steps before the system went extinct) for each of these settings, the following options were used for the simulation:
\begin{itemize}
\item Grid size of 100*100.
\item Birth Probability of 0.99 (based on the probability 1 used by Ruxton and Saravia, however altered so that it could be converted to a rate for the Gillespie simulator).
\item Death Probability of 0.6 (a value higher than the CP$_{e}$ for all settings (taken from Ruxton and Saravia)).
\item Initial population size of 1000 (10\% of the grid).
\item Run with a high enough value for the number of steps to allow the system to go extinct.
\end{itemize}
These simulations were run 10000 times, and the median values for number of steps taken before the system went extinct was taken as the Extinction Median.

\subsection{Persistent System}
\label{sec:exp.rux.pers}
For the simulations which fluctuate around an equilibrium value for the population, experiments were carried out to determine whether there are any differences in the systems for the different spatio-temporal rules. The main measure used for this is the Equilibrium Density (ED), i.e. the long-term temporal average of the number of sites occupied at a given time. This is done by running simulations with the following options:
\begin{itemize}
\item Grid size of 100*100.
\item Birth Probability of 0.99 (used for the same reason as the previous experiment).
\item Death Probability of 0.4 (value taken from Ruxton and Saravia's experiments).
\item Initial population size of 1000.
\item Run for 1500 steps (run for longer than Ruxton and Saravia's experiments as later systems required more time to settle).
\end{itemize}

The percentage of the grid that is occupied at each step is taken, then the average value for all of the steps are taken for the ED. This experiment was repeated 10 times and an average of the results was taken.

\subsection{Spatial Indices}
\label{sec:exp.rux.spat}
Additional measurements were also calculated in order to characterise spatial patterns. The measurements taken of these were the number of patches(NP) and the Largest Patch Index(LPI)\cite{McGarigal}. Since the cellular automata are not well mixed, i.e. events only take place within the neighbourhood of an occupied cell, short range correlations should become apparent. In order to measure such correlations, Moran's I spatial autocorrelation index (MI)\cite{Henebry} was used. This was done for all settings using a death probability of 0.4. Ruxton and Saravia also determined the Fractal Dimension of the cells, however this was not repeated in these experiments as it was decided that the patch measurements and Moran's I was enough to determine any differences in spatial behaviour.

The simulations are run using the following settings:
\begin{itemize}
\item Grid size of 100*100.
\item Birth Probability of 0.99.
\item Death Probability of 0.4.
\item Initial population size of 1000.
\item Run for 2000 steps.
\end{itemize}

The simulations were repeated 10 times and the averages were used as was done by Ruxton and Saravia.

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\section{Multiple Species and Birth and Death Probabilities}
\label{sec:exp.mult}
The experiments done by Ruxton and Saravia were all then repeated introducing a second species and varying the birth and death probabilities of each.
The Extinction Times experiments (mentioned in Section (\ref{sec:sec:exp.rux.ext}) were only repeated with the birth and death probabilities of both species being 0.99 and 0.6 respectively as it would not make sense to vary them. For the other two experiments, four different sets of birth and death probabilities were used:

\begin{itemize}
\item Set 1 - Birth: 0.99 and Death: 0.4
\item Set 2 - Birth: 0.25 and Death: 0.1
\item Set 3 - Birth: 0.0625 and Death: 0.025
\item Set 4 - Birth:0.015625 and Death: 0.00625
\end{itemize}

These values were chosen such that each probability set is a quarter of the one preceding it. 

Each species was given one of the probability sets with every combination being tested. Since giving species one and two the probability sets 1 and 2 respectively is the equivalent to giving them the probability sets 2 and 1 respectively, these combinations were not repeated. Therefore the number of actual combinations was equal to n(n+1)/2 where `n' is the number of different sets. So in this case, 10 different combinations were tested for each discrete step rule and the Gillespie simulator.

From the results of first set of experiments (see Section \ref{sec:anal.rux}), certain settings in the cellular automaton could be seen to be clearly paired together. As such, the experiments for this version (except for the Extinction Times one) were run with only one setting per pair was used. The ones chosen were SdF1, SdC1, RdF2 and RsR2. Although SdC1 and SdC2 showed more differences than the other pairs, it was decided that since the SdC2 setting tended to go extinct faster than any other setting, it would most likely not yield many usable results.


\subsection*{Summary}
This chapter explained all of the experiments that were performed and all of the parameters used for the simulator. It also explained what metrics were gathered from the experiments and how such that the reader should be able to recreate them. 
